MathematicsChapter 2: Quadratic Equations
USING THE FORMULA WHEN a IS NOT 1
Keep a in 2a. For 2x^2 + 3x - 2 = 0, D = 9 + 16 = 25.
Learning objectives
- Explain using the formula when a is not 1 using the correct definitions, conditions, and notation.
- Apply do not drop a, simplify +/- correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Keep a in 2a. For 2x2 + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match.
Mathematical form
a = 2, b = 3, c = -2
D = 32 - 4(2)(-2)
D = 9 + 16
D = 25
Where / symbols
- USING THE FORMULA WHEN a IS NOT 1
- Keep a in 2a. For 2x^2 + 3x - 2 = 0, D = 9 + 16 = 25. x = (-3 +/- 5)/4. That gives x=1/2 or x=-2. Factoring (2x-1)(x+2) should match.
- DISCRIMINANT
- D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.
- TWO ROOTS
- x = (-3+5)/4 = 2/4 = 1/2 and x = (-3-5)/4 = -8/4 = -2. Check in the original: 2(1/4)+3(1/2)-2=1/2+3/2-2=0.
Overview
Same formula, extra care with 2a:
Discriminant
D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.
Examples D=9-4(2)(-2)=25 · sqrt(25)=5 · Nice square
Two Roots
x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2. Check in the original: 2(14)+3(12)-2=12+32-2=0.
Examples x=1/2 · x=-2 · Check both
WHAT — USING THE FORMULA WHEN a IS NOT 1
Keep a in 2a. For 2x2 + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Using The Formula When A Is Not 1 matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through DISCRIMINANT, then TWO ROOTS. Worked: solve 2x^2 + 3x - 2 = 0. a = 2, b = 3, c = -2 D = 3^2 - 4(2)(-2) D = 9 + 16 D = 25 x = -3 +/- 54 x = 24 = 12 or x = -84 = -2 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- solve 2x2 + 3x - 2 = 0.
- a = 2, b = 3, c = -2
- D = 32 - 4(2)(-2)
- D = 9 + 16
- D = 25
- x = -3 +/- 54
- x = 24 = 12 or x = -84 = -2
In simple words
Denominator is 2a = 4, not 2. Forgetting a is the usual slip.
Key points
- do not drop a
- simplify +/-
- 4ac uses a
- two substitutions
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.USING THE FORMULA WHEN a IS NOT 1
Sticky Notes · Quick Revision
Core Idea
- Keep a in 2a. For 2x2 + 3x - 2 = 0, D = 9 + 16 = 25. x = -3 +/- 54. That gives x=12 or x=-2. Factoring (2x-1)(x+2) should match.
DISCRIMINANT
- D = b^2 - 4ac. The 4ac includes a. For 2x^2+3x-2, ac is negative, so -4ac is positive and D is larger.
TWO ROOTS
- x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2. Check in the original: 2(14)+3(12)-2=12+32-2=0.
Worked Method
- Worked: solve 2x2 + 3x - 2 = 0.
- a = 2, b = 3, c = -2
- D = 32 - 4(2)(-2)
- D = 9 + 16
- D = 25
- x = -3 +/- 54
- x = 24 = 12 or x = -84 = -2
Common Misconception
- Denominator is 2a = 4, not 2. Forgetting a is the usual slip.
FBISE Exam Check
- Denominator is 2a = 4, not 2. Forgetting a is the usual slip.
60-second revision
- Keep a in 2a.
- DISCRIMINANT: D = b2 - 4ac.
- TWO ROOTS: x = -3+54 = 24 = 12 and x = -3-54 = -84 = -2.
- Worked-method reminder: x = 24 = 12 or x = -84 = -2
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